scholarly journals Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise

2021 ◽  
Author(s):  
Carlo Marinelli ◽  
Lluís Quer-Sardanyons
Keyword(s):  
Absolute Continuity ◽  
Multiplicative Noise ◽  
Evolution Equations ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Stochastic Pde ◽  
Stochastic Evolution Equations ◽  
Lipschitz Continuous ◽  
Locally Lipschitz

AbstractWe prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in $\mathbb {R}^{d}$ ℝ d with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.

Limit measures and ergodicity of fractional stochastic reaction–diffusion equations on unbounded domains

2021 ◽  
pp. 2140012
Author(s):  
Zhang Chen ◽  
Bixiang Wang
Keyword(s):  
Invariant Measures ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Lipschitz Continuous ◽  
Bounded Interval ◽  
Locally Lipschitz ◽  
Stochastic Reaction ◽  
And Diffusion

This paper deals with invariant measures of fractional stochastic reaction–diffusion equations on unbounded domains with locally Lipschitz continuous drift and diffusion terms. We first prove the existence and regularity of invariant measures, and then show the tightness of the set of all invariant measures of the equation when the noise intensity varies in a bounded interval. We also prove that every limit of invariant measures of the perturbed systems is an invariant measure of the corresponding limiting system. Under further conditions, we establish the ergodicity and the exponentially mixing property of invariant measures.

scholarly journals Weak pullback mean random attractors for stochastic evolution equations and applications

2021 ◽  
pp. 2240001
Author(s):  
Anhui Gu
Keyword(s):  
Stochastic Models ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Random Attractors ◽  
Porous Media Equations ◽  
Stochastic Reaction

In this paper, we investigate the existence and uniqueness of weak pullback mean random attractors for abstract stochastic evolution equations with general diffusion terms in Bochner spaces. As applications, the existence and uniqueness of weak pullback mean random attractors for some stochastic models such as stochastic reaction–diffusion equations, the stochastic [Formula: see text]-Laplace equation and stochastic porous media equations are established.

scholarly journals Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities

2019 ◽  
Vol 40 (2) ◽  
pp. 1005-1050 ◽  
Author(s):  
Arnulf Jentzen ◽  
Primož Pušnik
Keyword(s):  
Strong Convergence ◽  
Approximation Method ◽  
Evolution Equations ◽  
Convergence Rates ◽  
Reaction Diffusion ◽  
Convergence Result ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Lipschitz Continuous ◽  
Strong Convergence Rates

Abstract In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. We establish strong convergence rates for this approximation method in the case of semilinear stochastic evolution equations with globally monotone coefficients. Our strong convergence result, in particular, applies to a class of stochastic reaction–diffusion partial differential equations.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

RANDOM ATTRACTORS OF STOCHASTIC REACTION–DIFFUSION EQUATIONS ON VARIABLE DOMAINS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 301-314 ◽  
Author(s):  
H. CRAUEL ◽  
P. E. KLOEDEN ◽  
MEIHUA YANG
Keyword(s):  
Stochastic Partial Differential Equation ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Diffusion Type ◽  
Variable Domains ◽  
Reference Domain ◽  
Stochastic Reaction

It is shown that a stochastic partial differential equation of the reaction–diffusion type on time-varying domains obtained by a temporally continuous dependent spatially diffeomorphic transformation of a reference domain, which is bounded with a smooth boundary, generates a "partial-random" dynamical system, which has a pathwise nonautonomous pullback attractor.

scholarly journals Solving Some Linear and Nonlinear PDEs Using Laplace Adomian Decomposition Method

2018 ◽  
Vol 1 (2) ◽  
pp. 9-31
Author(s):  
Attaullah
Keyword(s):  
Decomposition Method ◽  
Evolution Equations ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Reaction Diffusion ◽  
Nonlinear Evolution Equations ◽  
Reaction Diffusion Equations ◽  
Nonlinear Pdes ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

In this paper, Laplace Adomian decomposition method (LADM) is applied to solve linear and nonlinear partial differential equations (PDEs). With the help of proposed method, we handle the approximated analytical solutions to some interesting classes of PDEs including nonlinear evolution equations, Cauchy reaction-diffusion equations and the Klien-Gordon equations.

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